Complexity of a projected Newton-CG method for optimization with bounds

This paper describes a method for solving smooth nonconvex minimization problems subject to bound constraints with good worst-case complexity guarantees and practical performance. The method contains elements of two existing methods: the classical gradient projection approach for bound-constrained o...

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Bibliographic Details
Published in:Mathematical programming Vol. 207; no. 1-2; pp. 107 - 144
Main Authors: Xie, Yue, Wright, Stephen J.
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2024
Springer
Subjects:
Algorithms
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Comparative analysis
Full Length Paper
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Methods
Numerical Analysis
Theoretical
Conjugate gradient method
Complexity guarantees
68Q25
Projected gradient method
90C30
90C06
Newton’s method
49M15
90C60
Nonconvex bound-constrained optimization
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:This paper describes a method for solving smooth nonconvex minimization problems subject to bound constraints with good worst-case complexity guarantees and practical performance. The method contains elements of two existing methods: the classical gradient projection approach for bound-constrained optimization and a recently proposed Newton-conjugate gradient algorithm for unconstrained nonconvex optimization. Using a new definition of approximate second-order optimality parametrized by some tolerance ϵ (which is compared with related definitions from previous works), we derive complexity bounds in terms of ϵ for both the number of iterations required and the total amount of computation. The latter is measured by the number of gradient evaluations or Hessian-vector products. We also describe illustrative computational results on several test problems from low-rank matrix optimization.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-023-02000-z